This is actually to separate questions. Qualified candidates may receive funds from the federal Presidential Election Campaign Fund. Ballot access depends on the provisions of state laws. One does not imply the other.
Candidates who qualify may elect to receive federal funding from the Presidential Election Campaign Fund for either or both the primary and general elections. "Minor party" candidates are those whose party received at least 5% of the vote in the previous presidential election. "Major party" candidates are those whose part received at least 25% of the vote in the previous presidential election. Those who meet neither level are "new party" candidates. New party candidates may receive partial reimbursements after the election if they get at least 5% in the election.
In addition to the vote share, candidates must demonstrate having received at least minimum levels of contributions in "small" donations in each of 20 states or more to be eligible, and must agree to contribution and spending limits, and to making required disclosures. Many candidates who would be eligible elect not to agree to the limits, thus not accepting the funds. See this Wikipedia article (linked above in a comment) for more details.
Each US state determines what a candidate must do to be included on the ballot, and these rules vary. In many states any party whose candidate received more than a specified minimum percentage of the vote in a previous election will have its nominee included. In many cases the relevant election is the immediately previous election for Governor. Other candidates may typically qualify by obtaining signatures on a petition. In most states the signers must be registered voters, and a minimum number must come from each county or election district. The details vary by state.
For example, In Maryland, Code § 5-703 specifies 10,000 voters, or 1% of the votes cast for Governor in the previous election for Governor , whichever is less. However, under Code § 8-502 a candidate may be placed on the primary ballot by a petition signed by at least 400 registered voters in each Congressional district in the state, or if a candidate is "generally recognized" as a candidate for a "principal political party" as certified by the Secretary of State. Uncer code § 1-101 a "principal political party" us one of the two top vote-getters in the previous election for Governor. By the way, under code § 4-103 a political party, to retain that status, must poll at least 1% of the votes for the highest state-wide office in a general election, or have been selected as the registered choice of at least 1% of the registered voters.
All the above citations are to the Maryland election code. Other states will have roughly similar provisions.